§2.2 The sine and cosine functions

Introduction

The sine and cosine functions

Tangent lines drawn to $y = 2^x$ at $x = -2, 0, 2$ have slopes roughly $0.17$, $0.7$, and $2.8$. What pattern connects each slope to the function value?

On the graph of $y = \sin(x)$, the tangent lines at $x = 0$ and $x = \pm 2\pi$ have slope $1$, at $x = \pm\pi$ they have slope $-1$, and at $x = \pm\frac{\pi}{2}$ and $\pm\frac{3\pi}{2}$ they are horizontal.

The tangent slopes of $f(x) = \sin(x)$ read $1, 0, -1, 0, 1$ across $x = 0, \frac{\pi}{2}, \pi, \frac{3\pi}{2}, 2\pi$, and the limit definition gives $f'(0) = \lim_{h \to 0}\frac{\sin h}{h} = 1$. Plotting these slopes produces which familiar graph?

On the graph of $y = \cos(x)$, the tangent lines are horizontal at $x = -2\pi, 0, 2\pi$, have slope $-1$ at $x = \frac{\pi}{2}$, and have slope $1$ at $x = \frac{3\pi}{2}$.

The tangent slopes of $g(x) = \cos(x)$ read $0, -1, 0, 1, 0$ across $x = 0, \frac{\pi}{2}, \pi, \frac{3\pi}{2}, 2\pi$, and the limit definition gives $g'(\frac{\pi}{2}) = \lim_{h \to 0}\frac{\cos(\frac{\pi}{2}+h)}{h} = -1$. Which familiar graph do the slopes trace?

Which of the following is the derivative of the cosine function, that is $\frac{d}{dx}[\cos(x)]$?

Let $h(t) = 3\cos(t) - 4\sin(t)$. Which expression is $\frac{dh}{dt}$?

For $f(x) = 2x + \frac{\sin(x)}{2}$, compute the exact slope of the tangent line at $x = \frac{\pi}{6}$ as a decimal rounded to three places.

For $g(x) = x^2 + 2\cos(x)$ at $x = \frac{\pi}{2}$, we have $g(\frac{\pi}{2}) = \frac{\pi^2}{4}$ and $g'(\frac{\pi}{2}) = \pi - 2$. The tangent line is

For $p(z) = z^4 + 4^z + 4\cos(z) - \sin(\frac{\pi}{2})$, which expression is $p'(z)$?

For $p(z) = z^4 + 4^z + 4\cos(z) - \sin(\frac{\pi}{2})$, compute $p'(0)$ rounded to three decimals.

The island population $P(t) = 24 + 8\sin(t)$ is measured in hundreds with $t$ in decades since 2010. The quantity $P'(2) \approx -3.329$ tells us that on January 1, 2030 the population is

The slope of the tangent line to $y = \sin(x)$ at $x = 0$ equals $1$.